For a bilinear form \(\mathcal {A}\) , we write \(|\mathcal {A}|^2 = \text{trace}(\mathcal {A} {\mathcal {A}}^t)\, .\) 
Recall
Theorem F \((\) Weitzenböck formula\()\)  For every function \(f \in C^3(M)\, ,\)
\( \frac{1}{2}\Delta |\nabla f|^2 = | \operatorname{Hess}\, f|^2 + \langle \nabla \Delta f, \nabla f \rangle + \operatorname{Ric}\big (\nabla f,\nabla f\big ) \) 

Substituting the distance function \(r(x)\)  for \(f(x)\)  into Weitzenböck formula (REF ), we have inside the cut locus of \(x_0\)  \((|\nabla r| = 1)\) , via Gauss lemma
\( 0 = |\text{Hess}\, r|^2 + \frac{\partial }{\partial r} \Delta r + \text{Ric}(\frac{\partial }{\partial r}, \frac{\partial }{\partial r}) \) 
Since \(\operatorname{Hess} (\frac{\partial }{\partial r}, \frac{\partial }{\partial r}) = 0\, ,|\text{Hess}\, r|^2 \ge \frac{(\Delta r)^2}{n-1}\, ,\)  \(\text{Ric}(\frac{\partial }{\partial r}, \frac{\partial }{\partial r}) = \operatorname{Ric}^{\operatorname{M}}_{\operatorname{rad}}(r)\, \)  and (REF ) occurs, it follows that
\({\left\lbrace \begin{array}{ll}\frac{\partial }{\partial r} \big ( \Delta r \big ) + \frac{(\Delta r)^2}{n-1} + (n-1) G_1(r) \le 0 \quad \operatorname{on}\quad \overset{\circ }{B}_{t_1}(x_0) \\\Delta r = \frac{n-1}{r} + O(1)\quad \operatorname{as}\quad t \rightarrow 0^{+}\, .\end{array}\right.}\) 
Let \(g_1 = \Delta r\, ,\)  \(\kappa _1 = n-1 = \kappa _2\, .\)  Then (REF ) becomes (REF ) and (REF ) becomes (REF ). Via
(REF ) and (REF ) with \(G_i\)  satisfying (REF ) on \((0, t_1) \cap (0, t_2)\)  and \(\kappa _1\)  satisfying (REF ). Applying Theorem REF  we have (REF ) holds on \((0, t_1)\) , i.e., (REF ) holds in \(\overset{\circ }{B}_{t_1}(x_0)\, .\) 
If in addition, (REF ) occurs in \(D(x_0)\, ,\) 
then (REF ) holds pointwise on \(D(x_0)\) . By Lemma A or using Green's Identity and a double limiting argument
(see [1]},[2]}, [3]}), (REF ) holds weakly on \(M\) .
